The Most Valuable Thing in a Maths Lesson Might Be a Wrong Answer (Productive failure)

Here is a 100 word summary of a piece of research I found interesting and relevant to secondary/A level teachers. And three quick ideas to start applying tomorrow.

Article citation:

Kapur, M. (2011). A further study of productive failure in mathematical problem solving: unpacking the design components. Instructional Science, 39(4), 561–579. http://www.jstor.org/stable/23882881

100 word summary:

Manu Kapur’s study investigated Productive Failure, where students attempt complex mathematical problems before receiving formal instruction. Year 7 students tackling average-speed problems often failed to find correct solutions initially, yet they later outperformed students taught through direct instruction on procedural, conceptual, and transfer tasks. The Productive Failure students generated more diverse representations, including graphs, tables, and diagrams, which appeared to activate prior knowledge and prepare them to learn from subsequent teaching. The study suggests that carefully designed struggle, followed by explicit teacher-led consolidation, can produce deeper mathematical understanding than immediate instruction and guided practice alone.

My Three ideas to apply straight away:

1. Predict Before Teaching

Before introducing a new method give students 3-4 questions and ask:

"What approach might work here? Why?"

Collect methods, not answers. Then teach the formal technique. This creates a "time for telling" and prepares students to learn from the explanation.

2. Compare Wrong and Right Solutions

After a topic, display:

• a common incorrect solution

• a correct solution

Ask: "Where do these methods diverge?"

Research suggests the comparison is often more valuable than simply reviewing the correct answer.

3. Delay Help by Two Minutes

When students are stuck, avoid immediately demonstrating the method.

Instead ask:

• "What have you tried?"

• "Can you draw a diagram?"

• "Can you represent it another way?"

The goal is to generate representations and ideas before instruction.

Mantra: Attempt → Compare → Teach → Practise rather than Teach → Practise.

My final thoughts:

Don't ask whether students are getting answers right during the exploration phase. Ask whether they are generating, testing, and refining mathematical predictions.